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Theorem pwcfsdom 9365
Description: A corollary of Konig's Theorem konigth 9351. Theorem 11.28 of [TakeutiZaring] p. 108. (Contributed by Mario Carneiro, 20-Mar-2013.)
Hypothesis
Ref Expression
pwcfsdom.1 𝐻 = (𝑦 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓𝑦)))
Assertion
Ref Expression
pwcfsdom (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))
Distinct variable group:   𝐴,𝑓,𝑦
Allowed substitution hints:   𝐻(𝑦,𝑓)

Proof of Theorem pwcfsdom
Dummy variables 𝑤 𝑧 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 onzsl 7008 . . . 4 (𝐴 ∈ On ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
21biimpi 206 . . 3 (𝐴 ∈ On → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
3 cfom 9046 . . . . . . 7 (cf‘ω) = ω
4 aleph0 8849 . . . . . . . 8 (ℵ‘∅) = ω
54fveq2i 6161 . . . . . . 7 (cf‘(ℵ‘∅)) = (cf‘ω)
63, 5, 43eqtr4i 2653 . . . . . 6 (cf‘(ℵ‘∅)) = (ℵ‘∅)
7 fveq2 6158 . . . . . . 7 (𝐴 = ∅ → (ℵ‘𝐴) = (ℵ‘∅))
87fveq2d 6162 . . . . . 6 (𝐴 = ∅ → (cf‘(ℵ‘𝐴)) = (cf‘(ℵ‘∅)))
96, 8, 73eqtr4a 2681 . . . . 5 (𝐴 = ∅ → (cf‘(ℵ‘𝐴)) = (ℵ‘𝐴))
10 fvex 6168 . . . . . . . . 9 (ℵ‘𝐴) ∈ V
1110canth2 8073 . . . . . . . 8 (ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴)
1210pw2en 8027 . . . . . . . 8 𝒫 (ℵ‘𝐴) ≈ (2𝑜𝑚 (ℵ‘𝐴))
13 sdomentr 8054 . . . . . . . 8 (((ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴) ∧ 𝒫 (ℵ‘𝐴) ≈ (2𝑜𝑚 (ℵ‘𝐴))) → (ℵ‘𝐴) ≺ (2𝑜𝑚 (ℵ‘𝐴)))
1411, 12, 13mp2an 707 . . . . . . 7 (ℵ‘𝐴) ≺ (2𝑜𝑚 (ℵ‘𝐴))
15 alephon 8852 . . . . . . . . 9 (ℵ‘𝐴) ∈ On
16 alephgeom 8865 . . . . . . . . . 10 (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
17 omelon 8503 . . . . . . . . . . . 12 ω ∈ On
18 2onn 7680 . . . . . . . . . . . 12 2𝑜 ∈ ω
19 onelss 5735 . . . . . . . . . . . 12 (ω ∈ On → (2𝑜 ∈ ω → 2𝑜 ⊆ ω))
2017, 18, 19mp2 9 . . . . . . . . . . 11 2𝑜 ⊆ ω
21 sstr 3596 . . . . . . . . . . 11 ((2𝑜 ⊆ ω ∧ ω ⊆ (ℵ‘𝐴)) → 2𝑜 ⊆ (ℵ‘𝐴))
2220, 21mpan 705 . . . . . . . . . 10 (ω ⊆ (ℵ‘𝐴) → 2𝑜 ⊆ (ℵ‘𝐴))
2316, 22sylbi 207 . . . . . . . . 9 (𝐴 ∈ On → 2𝑜 ⊆ (ℵ‘𝐴))
24 ssdomg 7961 . . . . . . . . 9 ((ℵ‘𝐴) ∈ On → (2𝑜 ⊆ (ℵ‘𝐴) → 2𝑜 ≼ (ℵ‘𝐴)))
2515, 23, 24mpsyl 68 . . . . . . . 8 (𝐴 ∈ On → 2𝑜 ≼ (ℵ‘𝐴))
26 mapdom1 8085 . . . . . . . 8 (2𝑜 ≼ (ℵ‘𝐴) → (2𝑜𝑚 (ℵ‘𝐴)) ≼ ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴)))
2725, 26syl 17 . . . . . . 7 (𝐴 ∈ On → (2𝑜𝑚 (ℵ‘𝐴)) ≼ ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴)))
28 sdomdomtr 8053 . . . . . . 7 (((ℵ‘𝐴) ≺ (2𝑜𝑚 (ℵ‘𝐴)) ∧ (2𝑜𝑚 (ℵ‘𝐴)) ≼ ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴))) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴)))
2914, 27, 28sylancr 694 . . . . . 6 (𝐴 ∈ On → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴)))
30 oveq2 6623 . . . . . . 7 ((cf‘(ℵ‘𝐴)) = (ℵ‘𝐴) → ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))) = ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴)))
3130breq2d 4635 . . . . . 6 ((cf‘(ℵ‘𝐴)) = (ℵ‘𝐴) → ((ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))) ↔ (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴))))
3229, 31syl5ibrcom 237 . . . . 5 (𝐴 ∈ On → ((cf‘(ℵ‘𝐴)) = (ℵ‘𝐴) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))))
339, 32syl5 34 . . . 4 (𝐴 ∈ On → (𝐴 = ∅ → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))))
34 alephreg 9364 . . . . . . 7 (cf‘(ℵ‘suc 𝑥)) = (ℵ‘suc 𝑥)
35 fveq2 6158 . . . . . . . 8 (𝐴 = suc 𝑥 → (ℵ‘𝐴) = (ℵ‘suc 𝑥))
3635fveq2d 6162 . . . . . . 7 (𝐴 = suc 𝑥 → (cf‘(ℵ‘𝐴)) = (cf‘(ℵ‘suc 𝑥)))
3734, 36, 353eqtr4a 2681 . . . . . 6 (𝐴 = suc 𝑥 → (cf‘(ℵ‘𝐴)) = (ℵ‘𝐴))
3837rexlimivw 3024 . . . . 5 (∃𝑥 ∈ On 𝐴 = suc 𝑥 → (cf‘(ℵ‘𝐴)) = (ℵ‘𝐴))
3938, 32syl5 34 . . . 4 (𝐴 ∈ On → (∃𝑥 ∈ On 𝐴 = suc 𝑥 → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))))
40 cfsmo 9053 . . . . . 6 ((ℵ‘𝐴) ∈ On → ∃𝑓(𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤)))
41 limelon 5757 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ On)
42 ffn 6012 . . . . . . . . . . . . . . . 16 (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → 𝑓 Fn (cf‘(ℵ‘𝐴)))
43 fnrnfv 6209 . . . . . . . . . . . . . . . . 17 (𝑓 Fn (cf‘(ℵ‘𝐴)) → ran 𝑓 = {𝑦 ∣ ∃𝑥 ∈ (cf‘(ℵ‘𝐴))𝑦 = (𝑓𝑥)})
4443unieqd 4419 . . . . . . . . . . . . . . . 16 (𝑓 Fn (cf‘(ℵ‘𝐴)) → ran 𝑓 = {𝑦 ∣ ∃𝑥 ∈ (cf‘(ℵ‘𝐴))𝑦 = (𝑓𝑥)})
4542, 44syl 17 . . . . . . . . . . . . . . 15 (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ran 𝑓 = {𝑦 ∣ ∃𝑥 ∈ (cf‘(ℵ‘𝐴))𝑦 = (𝑓𝑥)})
46 fvex 6168 . . . . . . . . . . . . . . . 16 (𝑓𝑥) ∈ V
4746dfiun2 4527 . . . . . . . . . . . . . . 15 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓𝑥) = {𝑦 ∣ ∃𝑥 ∈ (cf‘(ℵ‘𝐴))𝑦 = (𝑓𝑥)}
4845, 47syl6eqr 2673 . . . . . . . . . . . . . 14 (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ran 𝑓 = 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓𝑥))
4948ad2antrl 763 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤))) → ran 𝑓 = 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓𝑥))
50 fnfvelrn 6322 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓 Fn (cf‘(ℵ‘𝐴)) ∧ 𝑤 ∈ (cf‘(ℵ‘𝐴))) → (𝑓𝑤) ∈ ran 𝑓)
5142, 50sylan 488 . . . . . . . . . . . . . . . . . . . 20 ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑤 ∈ (cf‘(ℵ‘𝐴))) → (𝑓𝑤) ∈ ran 𝑓)
52 sseq2 3612 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = (𝑓𝑤) → (𝑧𝑦𝑧 ⊆ (𝑓𝑤)))
5352rspcev 3299 . . . . . . . . . . . . . . . . . . . 20 (((𝑓𝑤) ∈ ran 𝑓𝑧 ⊆ (𝑓𝑤)) → ∃𝑦 ∈ ran 𝑓 𝑧𝑦)
5451, 53sylan 488 . . . . . . . . . . . . . . . . . . 19 (((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑤 ∈ (cf‘(ℵ‘𝐴))) ∧ 𝑧 ⊆ (𝑓𝑤)) → ∃𝑦 ∈ ran 𝑓 𝑧𝑦)
5554ex 450 . . . . . . . . . . . . . . . . . 18 ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑤 ∈ (cf‘(ℵ‘𝐴))) → (𝑧 ⊆ (𝑓𝑤) → ∃𝑦 ∈ ran 𝑓 𝑧𝑦))
5655rexlimdva 3026 . . . . . . . . . . . . . . . . 17 (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → (∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤) → ∃𝑦 ∈ ran 𝑓 𝑧𝑦))
5756ralimdv 2959 . . . . . . . . . . . . . . . 16 (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → (∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤) → ∀𝑧 ∈ (ℵ‘𝐴)∃𝑦 ∈ ran 𝑓 𝑧𝑦))
5857imp 445 . . . . . . . . . . . . . . 15 ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤)) → ∀𝑧 ∈ (ℵ‘𝐴)∃𝑦 ∈ ran 𝑓 𝑧𝑦)
5958adantl 482 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤))) → ∀𝑧 ∈ (ℵ‘𝐴)∃𝑦 ∈ ran 𝑓 𝑧𝑦)
60 alephislim 8866 . . . . . . . . . . . . . . . 16 (𝐴 ∈ On ↔ Lim (ℵ‘𝐴))
6160biimpi 206 . . . . . . . . . . . . . . 15 (𝐴 ∈ On → Lim (ℵ‘𝐴))
62 frn 6020 . . . . . . . . . . . . . . . 16 (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ran 𝑓 ⊆ (ℵ‘𝐴))
6362adantr 481 . . . . . . . . . . . . . . 15 ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤)) → ran 𝑓 ⊆ (ℵ‘𝐴))
64 coflim 9043 . . . . . . . . . . . . . . 15 ((Lim (ℵ‘𝐴) ∧ ran 𝑓 ⊆ (ℵ‘𝐴)) → ( ran 𝑓 = (ℵ‘𝐴) ↔ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑦 ∈ ran 𝑓 𝑧𝑦))
6561, 63, 64syl2an 494 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤))) → ( ran 𝑓 = (ℵ‘𝐴) ↔ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑦 ∈ ran 𝑓 𝑧𝑦))
6659, 65mpbird 247 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤))) → ran 𝑓 = (ℵ‘𝐴))
6749, 66eqtr3d 2657 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤))) → 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓𝑥) = (ℵ‘𝐴))
68 ffvelrn 6323 . . . . . . . . . . . . . . . . 17 ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝑓𝑥) ∈ (ℵ‘𝐴))
6915oneli 5804 . . . . . . . . . . . . . . . . 17 ((𝑓𝑥) ∈ (ℵ‘𝐴) → (𝑓𝑥) ∈ On)
70 harcard 8764 . . . . . . . . . . . . . . . . . . 19 (card‘(har‘(𝑓𝑥))) = (har‘(𝑓𝑥))
71 iscard 8761 . . . . . . . . . . . . . . . . . . . 20 ((card‘(har‘(𝑓𝑥))) = (har‘(𝑓𝑥)) ↔ ((har‘(𝑓𝑥)) ∈ On ∧ ∀𝑦 ∈ (har‘(𝑓𝑥))𝑦 ≺ (har‘(𝑓𝑥))))
7271simprbi 480 . . . . . . . . . . . . . . . . . . 19 ((card‘(har‘(𝑓𝑥))) = (har‘(𝑓𝑥)) → ∀𝑦 ∈ (har‘(𝑓𝑥))𝑦 ≺ (har‘(𝑓𝑥)))
7370, 72ax-mp 5 . . . . . . . . . . . . . . . . . 18 𝑦 ∈ (har‘(𝑓𝑥))𝑦 ≺ (har‘(𝑓𝑥))
74 domrefg 7950 . . . . . . . . . . . . . . . . . . . 20 ((𝑓𝑥) ∈ V → (𝑓𝑥) ≼ (𝑓𝑥))
7546, 74ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝑓𝑥) ≼ (𝑓𝑥)
76 elharval 8428 . . . . . . . . . . . . . . . . . . . 20 ((𝑓𝑥) ∈ (har‘(𝑓𝑥)) ↔ ((𝑓𝑥) ∈ On ∧ (𝑓𝑥) ≼ (𝑓𝑥)))
7776biimpri 218 . . . . . . . . . . . . . . . . . . 19 (((𝑓𝑥) ∈ On ∧ (𝑓𝑥) ≼ (𝑓𝑥)) → (𝑓𝑥) ∈ (har‘(𝑓𝑥)))
7875, 77mpan2 706 . . . . . . . . . . . . . . . . . 18 ((𝑓𝑥) ∈ On → (𝑓𝑥) ∈ (har‘(𝑓𝑥)))
79 breq1 4626 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (𝑓𝑥) → (𝑦 ≺ (har‘(𝑓𝑥)) ↔ (𝑓𝑥) ≺ (har‘(𝑓𝑥))))
8079rspccv 3296 . . . . . . . . . . . . . . . . . 18 (∀𝑦 ∈ (har‘(𝑓𝑥))𝑦 ≺ (har‘(𝑓𝑥)) → ((𝑓𝑥) ∈ (har‘(𝑓𝑥)) → (𝑓𝑥) ≺ (har‘(𝑓𝑥))))
8173, 78, 80mpsyl 68 . . . . . . . . . . . . . . . . 17 ((𝑓𝑥) ∈ On → (𝑓𝑥) ≺ (har‘(𝑓𝑥)))
8268, 69, 813syl 18 . . . . . . . . . . . . . . . 16 ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝑓𝑥) ≺ (har‘(𝑓𝑥)))
83 harcl 8426 . . . . . . . . . . . . . . . . . . 19 (har‘(𝑓𝑥)) ∈ On
84 fveq2 6158 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑥 → (𝑓𝑦) = (𝑓𝑥))
8584fveq2d 6162 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑥 → (har‘(𝑓𝑦)) = (har‘(𝑓𝑥)))
86 pwcfsdom.1 . . . . . . . . . . . . . . . . . . . 20 𝐻 = (𝑦 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓𝑦)))
8785, 86fvmptg 6247 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ (cf‘(ℵ‘𝐴)) ∧ (har‘(𝑓𝑥)) ∈ On) → (𝐻𝑥) = (har‘(𝑓𝑥)))
8883, 87mpan2 706 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (cf‘(ℵ‘𝐴)) → (𝐻𝑥) = (har‘(𝑓𝑥)))
8988breq2d 4635 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (cf‘(ℵ‘𝐴)) → ((𝑓𝑥) ≺ (𝐻𝑥) ↔ (𝑓𝑥) ≺ (har‘(𝑓𝑥))))
9089adantl 482 . . . . . . . . . . . . . . . 16 ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → ((𝑓𝑥) ≺ (𝐻𝑥) ↔ (𝑓𝑥) ≺ (har‘(𝑓𝑥))))
9182, 90mpbird 247 . . . . . . . . . . . . . . 15 ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝑓𝑥) ≺ (𝐻𝑥))
9291ralrimiva 2962 . . . . . . . . . . . . . 14 (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓𝑥) ≺ (𝐻𝑥))
93 fvex 6168 . . . . . . . . . . . . . . 15 (cf‘(ℵ‘𝐴)) ∈ V
94 eqid 2621 . . . . . . . . . . . . . . 15 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓𝑥) = 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓𝑥)
95 eqid 2621 . . . . . . . . . . . . . . 15 X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) = X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥)
9693, 94, 95konigth 9351 . . . . . . . . . . . . . 14 (∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓𝑥) ≺ (𝐻𝑥) → 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓𝑥) ≺ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥))
9792, 96syl 17 . . . . . . . . . . . . 13 (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓𝑥) ≺ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥))
9897ad2antrl 763 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤))) → 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓𝑥) ≺ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥))
9967, 98eqbrtrrd 4647 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤))) → (ℵ‘𝐴) ≺ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥))
10041, 99sylan 488 . . . . . . . . . 10 (((𝐴 ∈ V ∧ Lim 𝐴) ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤))) → (ℵ‘𝐴) ≺ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥))
101 ovex 6643 . . . . . . . . . . . 12 ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))) ∈ V
10268ex 450 . . . . . . . . . . . . . . . 16 (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → (𝑥 ∈ (cf‘(ℵ‘𝐴)) → (𝑓𝑥) ∈ (ℵ‘𝐴)))
103 alephlim 8850 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) = 𝑦𝐴 (ℵ‘𝑦))
104103eleq2d 2684 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ V ∧ Lim 𝐴) → ((𝑓𝑥) ∈ (ℵ‘𝐴) ↔ (𝑓𝑥) ∈ 𝑦𝐴 (ℵ‘𝑦)))
105 eliun 4497 . . . . . . . . . . . . . . . . . . 19 ((𝑓𝑥) ∈ 𝑦𝐴 (ℵ‘𝑦) ↔ ∃𝑦𝐴 (𝑓𝑥) ∈ (ℵ‘𝑦))
106 alephcard 8853 . . . . . . . . . . . . . . . . . . . . . . . . 25 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)
107106eleq2i 2690 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑓𝑥) ∈ (card‘(ℵ‘𝑦)) ↔ (𝑓𝑥) ∈ (ℵ‘𝑦))
108 cardsdomelir 8759 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑓𝑥) ∈ (card‘(ℵ‘𝑦)) → (𝑓𝑥) ≺ (ℵ‘𝑦))
109107, 108sylbir 225 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓𝑥) ∈ (ℵ‘𝑦) → (𝑓𝑥) ≺ (ℵ‘𝑦))
110 elharval 8428 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((ℵ‘𝑦) ∈ (har‘(𝑓𝑥)) ↔ ((ℵ‘𝑦) ∈ On ∧ (ℵ‘𝑦) ≼ (𝑓𝑥)))
111110simprbi 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((ℵ‘𝑦) ∈ (har‘(𝑓𝑥)) → (ℵ‘𝑦) ≼ (𝑓𝑥))
112 domnsym 8046 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((ℵ‘𝑦) ≼ (𝑓𝑥) → ¬ (𝑓𝑥) ≺ (ℵ‘𝑦))
113111, 112syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((ℵ‘𝑦) ∈ (har‘(𝑓𝑥)) → ¬ (𝑓𝑥) ≺ (ℵ‘𝑦))
114113con2i 134 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑓𝑥) ≺ (ℵ‘𝑦) → ¬ (ℵ‘𝑦) ∈ (har‘(𝑓𝑥)))
115 alephon 8852 . . . . . . . . . . . . . . . . . . . . . . . . 25 (ℵ‘𝑦) ∈ On
116 ontri1 5726 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((har‘(𝑓𝑥)) ∈ On ∧ (ℵ‘𝑦) ∈ On) → ((har‘(𝑓𝑥)) ⊆ (ℵ‘𝑦) ↔ ¬ (ℵ‘𝑦) ∈ (har‘(𝑓𝑥))))
11783, 115, 116mp2an 707 . . . . . . . . . . . . . . . . . . . . . . . 24 ((har‘(𝑓𝑥)) ⊆ (ℵ‘𝑦) ↔ ¬ (ℵ‘𝑦) ∈ (har‘(𝑓𝑥)))
118114, 117sylibr 224 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓𝑥) ≺ (ℵ‘𝑦) → (har‘(𝑓𝑥)) ⊆ (ℵ‘𝑦))
119109, 118syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑓𝑥) ∈ (ℵ‘𝑦) → (har‘(𝑓𝑥)) ⊆ (ℵ‘𝑦))
120 alephord2i 8860 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴 ∈ On → (𝑦𝐴 → (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
121120imp 445 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ On ∧ 𝑦𝐴) → (ℵ‘𝑦) ∈ (ℵ‘𝐴))
122 ontr2 5741 . . . . . . . . . . . . . . . . . . . . . . 23 (((har‘(𝑓𝑥)) ∈ On ∧ (ℵ‘𝐴) ∈ On) → (((har‘(𝑓𝑥)) ⊆ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ∈ (ℵ‘𝐴)) → (har‘(𝑓𝑥)) ∈ (ℵ‘𝐴)))
12383, 15, 122mp2an 707 . . . . . . . . . . . . . . . . . . . . . 22 (((har‘(𝑓𝑥)) ⊆ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ∈ (ℵ‘𝐴)) → (har‘(𝑓𝑥)) ∈ (ℵ‘𝐴))
124119, 121, 123syl2anr 495 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴 ∈ On ∧ 𝑦𝐴) ∧ (𝑓𝑥) ∈ (ℵ‘𝑦)) → (har‘(𝑓𝑥)) ∈ (ℵ‘𝐴))
125124exp31 629 . . . . . . . . . . . . . . . . . . . 20 (𝐴 ∈ On → (𝑦𝐴 → ((𝑓𝑥) ∈ (ℵ‘𝑦) → (har‘(𝑓𝑥)) ∈ (ℵ‘𝐴))))
126125rexlimdv 3025 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ On → (∃𝑦𝐴 (𝑓𝑥) ∈ (ℵ‘𝑦) → (har‘(𝑓𝑥)) ∈ (ℵ‘𝐴)))
127105, 126syl5bi 232 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ On → ((𝑓𝑥) ∈ 𝑦𝐴 (ℵ‘𝑦) → (har‘(𝑓𝑥)) ∈ (ℵ‘𝐴)))
12841, 127syl 17 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ V ∧ Lim 𝐴) → ((𝑓𝑥) ∈ 𝑦𝐴 (ℵ‘𝑦) → (har‘(𝑓𝑥)) ∈ (ℵ‘𝐴)))
129104, 128sylbid 230 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ V ∧ Lim 𝐴) → ((𝑓𝑥) ∈ (ℵ‘𝐴) → (har‘(𝑓𝑥)) ∈ (ℵ‘𝐴)))
130102, 129sylan9r 689 . . . . . . . . . . . . . . 15 (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) → (𝑥 ∈ (cf‘(ℵ‘𝐴)) → (har‘(𝑓𝑥)) ∈ (ℵ‘𝐴)))
131130imp 445 . . . . . . . . . . . . . 14 ((((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (har‘(𝑓𝑥)) ∈ (ℵ‘𝐴))
13285cbvmptv 4720 . . . . . . . . . . . . . . 15 (𝑦 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓𝑦))) = (𝑥 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓𝑥)))
13386, 132eqtri 2643 . . . . . . . . . . . . . 14 𝐻 = (𝑥 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓𝑥)))
134131, 133fmptd 6351 . . . . . . . . . . . . 13 (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) → 𝐻:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴))
135 ffvelrn 6323 . . . . . . . . . . . . . . 15 ((𝐻:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝐻𝑥) ∈ (ℵ‘𝐴))
136 onelss 5735 . . . . . . . . . . . . . . 15 ((ℵ‘𝐴) ∈ On → ((𝐻𝑥) ∈ (ℵ‘𝐴) → (𝐻𝑥) ⊆ (ℵ‘𝐴)))
13715, 135, 136mpsyl 68 . . . . . . . . . . . . . 14 ((𝐻:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝐻𝑥) ⊆ (ℵ‘𝐴))
138137ralrimiva 2962 . . . . . . . . . . . . 13 (𝐻:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) ⊆ (ℵ‘𝐴))
139 ss2ixp 7881 . . . . . . . . . . . . . 14 (∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) ⊆ (ℵ‘𝐴) → X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) ⊆ X𝑥 ∈ (cf‘(ℵ‘𝐴))(ℵ‘𝐴))
14093, 10ixpconst 7878 . . . . . . . . . . . . . 14 X𝑥 ∈ (cf‘(ℵ‘𝐴))(ℵ‘𝐴) = ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))
141139, 140syl6sseq 3636 . . . . . . . . . . . . 13 (∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) ⊆ (ℵ‘𝐴) → X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) ⊆ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
142134, 138, 1413syl 18 . . . . . . . . . . . 12 (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) → X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) ⊆ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
143 ssdomg 7961 . . . . . . . . . . . 12 (((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))) ∈ V → (X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) ⊆ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))) → X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) ≼ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))))
144101, 142, 143mpsyl 68 . . . . . . . . . . 11 (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) → X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) ≼ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
145144adantrr 752 . . . . . . . . . 10 (((𝐴 ∈ V ∧ Lim 𝐴) ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤))) → X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) ≼ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
146 sdomdomtr 8053 . . . . . . . . . 10 (((ℵ‘𝐴) ≺ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) ∧ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) ≼ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
147100, 145, 146syl2anc 692 . . . . . . . . 9 (((𝐴 ∈ V ∧ Lim 𝐴) ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤))) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
148147expcom 451 . . . . . . . 8 ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤)) → ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))))
1491483adant2 1078 . . . . . . 7 ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤)) → ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))))
150149exlimiv 1855 . . . . . 6 (∃𝑓(𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤)) → ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))))
15115, 40, 150mp2b 10 . . . . 5 ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
152151a1i 11 . . . 4 (𝐴 ∈ On → ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))))
15333, 39, 1523jaod 1389 . . 3 (𝐴 ∈ On → ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))))
1542, 153mpd 15 . 2 (𝐴 ∈ On → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
155 alephfnon 8848 . . . . 5 ℵ Fn On
156 fndm 5958 . . . . 5 (ℵ Fn On → dom ℵ = On)
157155, 156ax-mp 5 . . . 4 dom ℵ = On
158157eleq2i 2690 . . 3 (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
159 ndmfv 6185 . . . 4 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
160 1n0 7535 . . . . . 6 1𝑜 ≠ ∅
161 1on 7527 . . . . . . . 8 1𝑜 ∈ On
162161elexi 3203 . . . . . . 7 1𝑜 ∈ V
1631620sdom 8051 . . . . . 6 (∅ ≺ 1𝑜 ↔ 1𝑜 ≠ ∅)
164160, 163mpbir 221 . . . . 5 ∅ ≺ 1𝑜
165 id 22 . . . . . 6 ((ℵ‘𝐴) = ∅ → (ℵ‘𝐴) = ∅)
166 fveq2 6158 . . . . . . . . 9 ((ℵ‘𝐴) = ∅ → (cf‘(ℵ‘𝐴)) = (cf‘∅))
167 cf0 9033 . . . . . . . . 9 (cf‘∅) = ∅
168166, 167syl6eq 2671 . . . . . . . 8 ((ℵ‘𝐴) = ∅ → (cf‘(ℵ‘𝐴)) = ∅)
169165, 168oveq12d 6633 . . . . . . 7 ((ℵ‘𝐴) = ∅ → ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))) = (∅ ↑𝑚 ∅))
170 0ex 4760 . . . . . . . 8 ∅ ∈ V
171 map0e 7855 . . . . . . . 8 (∅ ∈ V → (∅ ↑𝑚 ∅) = 1𝑜)
172170, 171ax-mp 5 . . . . . . 7 (∅ ↑𝑚 ∅) = 1𝑜
173169, 172syl6eq 2671 . . . . . 6 ((ℵ‘𝐴) = ∅ → ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))) = 1𝑜)
174165, 173breq12d 4636 . . . . 5 ((ℵ‘𝐴) = ∅ → ((ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))) ↔ ∅ ≺ 1𝑜))
175164, 174mpbiri 248 . . . 4 ((ℵ‘𝐴) = ∅ → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
176159, 175syl 17 . . 3 𝐴 ∈ dom ℵ → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
177158, 176sylnbir 321 . 2 𝐴 ∈ On → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
178154, 177pm2.61i 176 1 (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3o 1035  w3a 1036   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wne 2790  wral 2908  wrex 2909  Vcvv 3190  wss 3560  c0 3897  𝒫 cpw 4136   cuni 4409   ciun 4492   class class class wbr 4623  cmpt 4683  dom cdm 5084  ran crn 5085  Oncon0 5692  Lim wlim 5693  suc csuc 5694   Fn wfn 5852  wf 5853  cfv 5857  (class class class)co 6615  ωcom 7027  Smo wsmo 7402  1𝑜c1o 7513  2𝑜c2o 7514  𝑚 cmap 7817  Xcixp 7868  cen 7912  cdom 7913  csdm 7914  harchar 8421  cardccrd 8721  cale 8722  cfccf 8723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498  ax-ac2 9245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-iin 4495  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-smo 7403  df-recs 7428  df-rdg 7466  df-1o 7520  df-2o 7521  df-oadd 7524  df-er 7702  df-map 7819  df-ixp 7869  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-oi 8375  df-har 8423  df-card 8725  df-aleph 8726  df-cf 8727  df-acn 8728  df-ac 8899
This theorem is referenced by:  cfpwsdom  9366  tskcard  9563  bj-pwcfsdom  32722
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